A Robust FocusedandDeconvolved Conventional Beamforming for a Uniform Linear Array
https://doi.org/10.1007/s11804024004252

Abstract
In the field of array signal processing,uniform linear arrays (ULAs) are widely used to detect/separate a weak target and estimate its direction of arrival from interference and noise. Conventional beamforming (CBF) is robust but restricted by a wide mainlobe and high sidelobe level. Covariancematrixinversed beamforming techniques,such as the minimum variance distortionless response and multiple signal classification,are sensitive to signal mismatch and data snapshots and exhibit highresolution performance because of the narrow mainlobe and low sidelobe level. Therefore,compared with the wideband CBF,this study proposes a robust focusedanddeconvolved conventional beamforming (RFDCBF),utilizing the Richardson – Lucy (RL) iterative algorithm to deconvolve the focused conventional beam power of a halfwavelength spaced ULA. Then,the focusedanddeconvolved beam power achieves a narrower mainlobe and lower sidelobe level while retaining the robustness of wideband CBF. Moreover,compared with the wideband CBF,RFDCBF can obtain a higher output signaltonoise ratio (SNR). Finally,the performance of RFDCBF is evaluated through numerical simulation and verified by sea trial data processing.
Keywords:
 Wideband ·
 Beamforming ·
 Focusing transform ·
 Deconvolution ·
 High resolution ·
 Robust
Article Highlights● Based on the focusing transform, we extend the deconvolved conventional beamforming from narrowband to wideband.● The wideband focusedanddeconvolved beam power yields a narrower mainlobe and lower sidelobe level than the wideband conventional beam power.● The feasibility of practical application is verified by numerical simulation and realdata processing results. 
1 Introduction
In the field of array signal processing (Zhang et al., 2020; Sheng et al., 2023; Lu, 2023), beamforming techniques have been playing a key role in underwater acoustic engineering. Among the beamforming algorithms, the conventional beamforming (CBF) has been widely applied in underwater acoustic array signal processing because of its robustness against signal mismatch, and it requires only a few data snapshots to achieve a reliable estimation of target angles. However, CBF still has some disadvantages, such as the wide mainlobe (making it difficult to detect two neighboring targets of equal strength) and the high sidelobe level (making it difficult to detect weak targets with loud interference).
The decomposition of the received wideband data into many narrowband data or nonoverlapping bandpass filter banks in the frequency domain, followed by the existing narrowband beamforming algorithms, is an intuitive method of wideband signal processing. Among the beamforming algorithms, the easiest way to process wideband signals is the incoherent signal subspace method (ISSM) (Zhang et al., 2010; Ahmad et al., 2018), and the output beam power can be obtained using the average of all frequency subbands. Although ISSM can be applicable at different SNRs, its overall performance is inferior because of its poor beamforming at certain frequency bands (Hu et al., 2018). Subsequently, the coherent signal subspace method (CSSM) is utilized to obtain a set of narrowband signals via multiple narrowband filter banks in the frequency domain (Wang and Kaveh, 1985; Chen and Zhao, 2005; Li et al., 2018). For a specified angle, CSSM aligns the array flow patterns of each frequency dot to a uniform array flow pattern at the focusing frequency. Referring to a previous study (Swingler and Krolik, 1989), the estimation deviation of the CSSM method increases with the increase in bandwidth. Moreover, the output SNR changes after the focusing transform. By contrast, the SNR does not suffer a loss by the unitary focusing transform. Thus, Hung and Kaveh (1988) proposed the rotational signal subspace (RSS) method, which designs the unitary focusing transform matrix by presetting the steering vector of the target angle. Ma and Zhang (2019) introduced a method that can minimize the focusing transform error. Sellone (2006) proposed a robust CSSM (RCSM) based focusing transform that does not require prior knowledge of target angles and decreases the focusing transform error by iteratively narrowing the bearing interval between targets. Furthermore, the iterative process results in a significant increase in the calculation amount.
Yang(2017, 2018) applied the deconvolution method to CBF and proposed a deconvolved CBF algorithm that can reduce the mainlobe width, lowers the sidelobe level, and retains the robustness of CBF (Bahr and Cattafesta, 2012). Recently, global experts have focused considerable attention on the actual application of deconvolution beamforming algorithms, such as uniform acoustic pressure array (Xenaki et al., 2012), uniform circular array (TianaRoig and Jacobsen, 2013), vector array (Sun et al., 2019), and acoustic image measurement (Mei, 2020). As a result, the advantages of the deconvolution beamforming algorithm, such as high bearing resolution, low sidelobe level, and robustness, have been verified.
Therefore, for wideband CBF, by combining focusing transform with deconvolution, a focusedanddeconvolved beamforming algorithm is proposed. First, wideband CBF is converted into narrowband CBF at the focusing frequency by the focusing transform. Then, a highresolution beam power is achieved, deconvolving the wideband beam power. As a result, the focusedanddeconvolved beam power is confirmed to yield a narrower mainlobe and lower sidelobe level and retain the robustness of wideband CBF.
The outline of this paper is organized as follows: First, Section 2 introduces the basic principle, which is composed of three parts: (a) signal modeling; (b) the focusing transform methods, i.e., RSS and RCSM; and (c) the Richardson–Lucy (RL) iterative algorithm. Then, Section 3 presents the numerical simulation results of the output beam power, the high bearing resolution, and the antinoise performance. Subsequently, using the sea trial data processing results, Section 4 sufficiently verifies the feasibility of the proposed algorithm. Finally, Section 5 provides the conclusion of this paper.
2 Basic principle
2.1 Signal modeling
Assuming that, under farfield conditions, an Nsensor uniform linear array (ULA) is utilized for direction finding. With the frequency band [f_{L}, f_{H}] divided into J subbands, the array output of the j^{th} subband is derived as follows:
$$ \boldsymbol{X}\left(f_j\right)=\boldsymbol{A}\left(f_j\right) \boldsymbol{S}\left(f_j\right)+\boldsymbol{N}\left(f_j\right) $$ (1) where f_{j} is the center frequency of the j^{th} subband, A(f_{j}) is the array flow pattern, S (f_{j}) is the source signal, and N (f_{j}) is the Gaussian white noise, uncorrelated to the source.
First, the focusing transform matrix T (f_{j}) satisfies the Eq. (4) as follows:
$$ \boldsymbol{T}\left(f_j\right) \boldsymbol{A}\left(f_j\right)=\boldsymbol{A}\left(f_0\right), j=1, \cdots, J $$ (2) where T (f_{j}) is the focusing transform matrix of the j^{th} subband, f_{0} is the focusing frequency dot, and A(f_{0}) is the array flow pattern.
Then, the array output X (f_{j}) is preprocessed by the formulated T (f_{j}), and the processing results are expressed as follows:
$$ \begin{aligned} \boldsymbol{T}\left(f_j\right) \boldsymbol{X}\left(f_j\right) & =\boldsymbol{T}\left(f_j\right) \boldsymbol{A}\left(f_j\right) \boldsymbol{S}\left(f_j\right)+\boldsymbol{T}\left(f_j\right) \boldsymbol{N}\left(f_j\right) \\ & =\boldsymbol{A}\left(f_0\right) \boldsymbol{S}\left(f_j\right)+\boldsymbol{T}\left(f_j\right) \boldsymbol{N}\left(f_j\right) \end{aligned} $$ (3) Subsequently, the CM of the received data is the sum and average of the CMs of all subbands, expressed as follows:
$$ \begin{aligned} \boldsymbol{R}_y & =\frac{1}{J} \sum\limits_{j=1}^J \boldsymbol{T}\left(f_j\right) \boldsymbol{X}\left(f_j\right) \boldsymbol{X}^{\mathrm{H}}\left(f_j\right) \boldsymbol{T}^{\mathrm{H}}\left(f_j\right) \\ & =\boldsymbol{A}\left(f_0\right)\left[\frac{1}{J} \sum\limits_{j=1}^J \boldsymbol{S}\left(f_j\right) \boldsymbol{S}^{\mathrm{H}}\left(f_j\right)\right] \boldsymbol{A}^{\mathrm{H}}\left(f_0\right) \\ & +\frac{1}{J} \sum\limits_{j=1}^J \boldsymbol{T}\left(f_j\right) \boldsymbol{N}\left(f_j\right) \boldsymbol{N}^{\mathrm{H}}\left(f_j\right) \boldsymbol{T}^{\mathrm{H}}\left(f_j\right) \end{aligned} $$ (4) Finally, the focused output beam power is derived as follows:
$$ \boldsymbol{P}_{\mathrm{CBF}}=\boldsymbol{W}^{\mathrm{H}} \boldsymbol{R}_y \boldsymbol{W} $$ (5) where W (θ) = [ω_{1}(θ), ω_{2}(θ), …, ω_{N}(θ)]^{H}.
2.2 RSSbased focusing transform
In this subsection, the optimal focusing transform matrix is formulated based on the minimum error criterion, expressed as follows:
$$ \left\{\begin{array}{l} \min \left\\boldsymbol{A}\left(f_0, \theta\right)\boldsymbol{T}\left(f_j\right) \boldsymbol{A}\left(f_j, \theta\right)\right\_{\mathrm{F}}^2 \\ \boldsymbol{T}^{\mathrm{H}}\left(f_j\right) \boldsymbol{T}\left(f_j\right)=\boldsymbol{I}, j=1, 2, \cdots, J \end{array}\right. $$ (6) where θ is the scanning angle; A(f_{j}, θ) and A(f_{0}, θ) are the array flow patterns of the j^{th} subband and the focusing frequency, respectively; I is a J × J identity matrix.
The output beam power of the focused CBF (FCBF) can be obtained by substituting T (f_{j}) into Eqs. (4) and (5).
2.3 RCSMbased focusing transform
Based on the RSSbased method, the output beam power, with prior knowledge of target angles, is obtained. However, in practice, it is infeasible for realtime data processing. Thus, a novel focusing transform method without prior knowledge of target angles, i. e., the RCSM method, is adopted.
$$ \begin{gathered} \boldsymbol{T}\left(f_j\right)=\arg \left\{\min _{\boldsymbol{T}} \int_{\frac{1}{2}}^{\frac{1}{2}}\left\\mathbf{T A}\left(u, f_j\right)\boldsymbol{A}\left(u, f_0\right)\right\_{\mathrm{F}}^2 \times \omega(u) \mathrm{d} u\right\} \\ \text { subject to } \boldsymbol{T}^{\mathrm{H}} \boldsymbol{T}=\boldsymbol{I}, j=1, 2, \cdots, J \end{gathered} $$ (7) where u = sin θ/2 is the scanning angle in sine form; A(u, f_{j}) and A(u, f_{0}) are the array flow patterns of the j^{th} subband and the focusing frequency, respectively; and I is a J × J identity matrix.
Based on the RCSMbased method, the output beam power of the robust focused conventional beamforming (RFCBF) can be obtained by substituting T(f_{j}) into Eqs. (4) and (5).
2.4 RL iterative algorithm
Based on the maximum likelihood of the Bayesian theory (Richardson, 1972), the RL iterative algorithm was proposed by Richardson and Lucy. A blurred image is deconvolved by a given point scattering function (PSF) according to the statistical criterion of Poisson noise. After a certain iteration number, the maximum likelihood estimation with high accuracy is obtained. Then, the relevant formula derivation (Ströhl and Kaminski, 2015; Ma et al., 2020) is briefly introduced.
Let us consider there exists a source signal s (x) that spreads through a channel whose impulse response function is h (yx). If n (y) is the isotropic noise, then the received signal r (y) can be written as follows:
$$ r(y)=h(y \mid x) * s(x)+n(y) $$ (8) Deconvolution is used to restore the source signal from the received data, given that the CIR is known. Assuming that both the received data and the CIR are positive definite functions, the source signal is restored using the RL iterative algorithm as follows:
$$ s^{(i+1)}(x)=s^{(i)}(x) \int\limits_{\infty}^{+\infty} \frac{h(y \mid x)}{r^{(i)}} r(y) \mathrm{d} y $$ (9) where both $r^{(i)}(y)=\int\limits_{\infty}^{+\infty} h(y \mid x) s^{(i)}(x) \mathrm{d} x$ and s^{(i)}(x) converge to a unique solution after I iterations, expressed as follows:
$$ \lim\limits_{i \rightarrow \infty} s^{(i)}(x)=\underset{s(x)}{\arg \min } L\left(r(y), \int\limits_{\infty}^{+\infty} h(y \mid x) s(x) \mathrm{d} x\right) $$ (10) where L (·) is Csiszar’s discrimination, which can be expressed as follows:
$$ \begin{aligned} & L(p(x), q(x)) \\ & \quad=\int\limits_{\infty}^{+\infty} p(x) \log \frac{p(x)}{q(x)} \mathrm{d} x\int\limits_{\infty}^{+\infty}[p(x)q(x)] \mathrm{d} x \end{aligned} $$ (11) where h (yx) is regarded as the PSF and assumed to be shiftinvariant and can be expressed as follows:
$$ h(y \mid x)=h(yx) $$ (12) The output beam power of CBF is regarded as the convolution between the array directivity pattern and the angle–amplitude distribution function of the sources. Thus, Eq. (5) is rewritten as follows:
$$ B_{\mathrm{CBF}}(\sin \theta)=\int_{1}^1 \mathrm{PSF}(\sin \theta\sin \mu) S(\sin \mu) \mathrm{d} \mu $$ (13) Given that the PSF is known, the angle–amplitude distribution function of the sources can be obtained by deconvolving the CBF beam power B_{CBF}.
2.5 Procedures of the focusedanddeconvolved method
Eqs. (10) and (13) show that the output beam power of FCBF and RFCBF can be separately deconvolved by the RL iterative algorithm. Therefore, as shown in Figure 1, two focusedanddeconvolved beamforming algorithms are proposed, i.e., FDCBF and RFDCBF.
3 Numerical simulation
A 32element halfwavelength spaced ULA is used to receive data. The target with an SNR of 25 dB consists of 50 consecutive 0.2 s chirp signals and arrives at the direction of arrival (DOA) of 30°. The ambient noise is Gaussian white noise, which is uncorrelated to the target. The operating frequency band ranges from 900 to 1 000 Hz. Notation: the sampling rate is 32 kHz; the focusing frequency is 1 kHz; the beam power is plotted in sine form; the observation angle measured from the broadside of the ULA ranges from −90° to 90°.
3.1 Output beam power
Figure 2(a) shows the output beam power of FDCBF compared with that of the RSSbased focusing transform, and Figure 2(b) shows the output beam power of RFDCBF compared with that of the RCSMbased focusing transform. The focusedanddeconvolved beam power has a narrower mainlobe and lower sidelobe level compared with the CBF beam power. Moreover, the background level of the focusedanddeconvolved beam power (FDCBF and RFDCBF) is 3 dB lower than those of conventional beam power (CBF) and focused beam power (FCBF and RFCBF).
Subsequently, the focusedanddeconvolved output beam power (FDCBF and RFDCBF) is analyzed by conducting 100 Monte Carlo trials of the mainlobe width, first sidelobe level, and root–mean–square error (RMSE).
3.1.1 Mainlobe width
Here, the mainlobe width is defined as the bearing interval between the first zero points of both sides of the main beam (Farina, 2002). For one target at 0°, the mainlobe width is analyzed with the number of elements ranging from 10 to 90. The other parameters are kept constant.
Figure 3 shows that the mainlobe width decreases with the increase in the number of elements. The mainlobe widths of CBF, FCBF, and RFCBF are close to each other, indicating that the focusing transform does not influence the mainlobe width. Furthermore, the mainlobe widths of FDCBF and RFDCBF become narrower. When the number of elements is 10, the mainlobe width of RFDCBF is wider than that of FDCBF. When the number of elements reaches 60 or more, the mainlobe width curves of FDCBF and RFDCBF coincide. Thus, we have proven that the focusedanddeconvolved method can narrow the mainlobe width.
3.1.2 First sidelobe level
For one target at 0°, the first sidelobe level is evaluated with the number of elements increasing. The other parameters are kept constant.
Figure 4 shows that the first sidelobe level decreases with the increase in the number of elements. With the number of elements increasing from 10 to 90, the first sidelobe levels of CBF, FCBF, and RFCBF nearly coincide, i. e., first decreasing from −12 dB to −13 dB and then keeping constant. Comparatively, the first sidelobe levels of FDCBF and RFDCBF are lower than that of FDCBF, and the first sidelobe level of RFDCBF is slightly higher than that of FDCBF. The first sidelobe level of FDCBF decreases from −20.5 dB to −28.1 dB, whereas the first sidelobe level of RFDCBF decreases from −20.1 dB to −26.8 dB. Thus, the focusedanddeconvolved method is confirmed to suppress the first sidelobe level.
3.1.3 Root–mean–square error
For one target at 0°, the RMSE of DOA estimation is analyzed. The other parameters are kept constant.
Figure 5(a) shows that the RMSE curves decrease from 0.18 to 0 with the increase in the SNR from −21 dB to 15 dB. Figure 5(b) shows that the RMSE curves decrease from 0.032 to 0 with the increase in the SNR from 0 dB to 15 dB. Moreover, the RMSEs of the FDCBF and RFDCBF are lower than those of the FCBF and RFCBF, indicating that the focusedanddeconvolved method can well enhance the RMSE of DOA estimation.
3.2 DOA estimation
Two targets of equal strength (0 dB) are used to analyze the bearing resolution performance.
By changing the bearing interval of two targets, Figures 6(a) and 6(b) show that the bearing resolution of FDCBF is 1.1°, and Figures 6(c) and 6(d) show that the bearing resolution of RFDCBF is 1.2°. Therefore, we have proven that compared with CBF, the focusedanddeconvolved method can enhance the bearing resolution performance.
To further investigate the bearing resolution performance, Figure 7 shows the bearing time records (BTRs) of CBF, FDCBF, and RFDCBF. The dynamic target (1) moves from −35° to −30°, with an SNR of −5 dB. The static targets (2 and 3) are kept constant at 32° and 33.3°, with SNRs of 0 and −25 dB, respectively.
Figure 7(a) shows that a dynamic target (1) and a static target (2) exist in the BTR of CBF. By contrast, Figures 7(b) and 7(c) show that a weak target (3), which is obscured by the sidelobe level of the static target (2), can be detected. Therefore, the focusedanddeconvolved CBFs (FDCBF and RFDCBF) gets proved to suppress the sidelobe level and detect the weak target.
3.3 Antinoise performance
Here, the average background level (ABL) (Sheng et al., 2020) and spatial spectrum gain (SSG) (Liu, 2021) are utilized to evaluate the antinoise performance.
3.3.1 Average background level
Figures 8(a) and 8(b) show that the ABL decreases with the SNRs of [−30 dB, 15 dB] and [0 dB, 15 dB], respectively. Thus, we have proven that the ABL curves of CBF, FCBF, and RFCBF coincide and converge to −25 dB, whereas the ABL curves of FDCBF and RFDCBF are lower, reaching −38 and −40 dB, respectively. In summary, the focusedanddeconvolved method can suppress the ABL.
3.3.2 Spatial spectrum gain
The SSG is defined as follows:
$$ \begin{aligned} \mathrm{SSG} & =101 \mathrm{~g}\left(2 / \int\limits_0^\pi P(\theta) \sin \theta \mathrm{d} \theta\right) \\ & 101 \mathrm{~g}\left(2 / \int\limits_0^\pi P_{\mathrm{CBF}}(\theta) \sin \theta \mathrm{d} \theta\right) \\ & =101 \mathrm{~g}\left(\int\limits_0^\pi P_{\mathrm{CBF}}(\theta) \sin \theta \mathrm{d} \theta / \int\limits_0^\pi P(\theta) \sin \theta \mathrm{d} \theta\right) \end{aligned} $$ (14) where P(θ) and P_{CBF}(θ) are the spatial spectra of a certain beamformer and CBF, respectively, and θ ∈ (0, π) is the scanning angle.
Figures 9(a) and 9(b) show that the SSG increases with the SNRs of [−30 dB, 15 dB] and [0 dB, 15 dB]. The SSGs of FCBF and RFCBF are kept constant at 0 dB, indicating that the focusing transform does not affect the output SNR. The SSG curve of FDCBF (RFDCBF) increases from 0.5 dB to 3.2 dB (3 dB), indicating that deconvolution causes an increase in the output SNR. Thus, we have proven that the focusedanddeconvolved method can increase the output SNR, thereby enhancing the antinoise performance.
4 Data processing
A sea trial of underwater target detection was conducted in the South China Sea, approximately 100 km far from the coastline of Sanya City, on September 24 and 25, 2020.
The entire sea trial included a launching vessel with a transmitting sound source, a receiving vessel with a horizontal ULA, and a reserved noncooperative vessel. A lowfrequency transducer (Model UW350) was selected as the transmitting sound source, with the source level set as 150 dB. A 128element ULA operating at 120 Hz was used to collect data. Noncooperative targets, such as fishing boats, appear occasionally but do not affect the trial.
4.1 High bearing resolution performance
Figure 10 shows the BTRs of CBF, FDCBF, and RFDCBF processed using the sea trial data of 100 s.
Compared with Figure 10(a), Figure 10(b) shows that FDCBF can spatially distinguish two neighboring targets, i. e., 1 and 2, with prior knowledge of target angles. By contrast, Figure 10(c) shows that RFDCBF can spatially distinguish two adjacent targets, i.e., 1 and 2, without prior knowledge of target angles. However, a few false targets are detected in Figure 10(b), although their background levels are slightly lower than those detected in Figure 10(c).
Thus, we have proven that the focusedanddeconvolved method can enhance the bearing resolution performance and suppress the background level to some extent.
4.2 Weak target detection
Figure 11 shows the BTRs of CBF, FDCBF, and RFDCBF processed using the sea trial data of 100 s.
Figure 11 shows two weak targets, i.e., 3 and 5, where the strength of Target 5 is slightly lower than that of Target 3. As shown in Figure 11(a), Target 3 appears from 40 to 100 s and disappears from 0 to 40 s, which can be attributed to the fact that the weak target (3) is obscured by the sidelobe level of the static target (2). Similarly, when the strength of Target 5 is low and even close to the background level, Target 5 appears from 0 to 10 s and disappears from 10 to 100 s. As shown in Figure 11(b), with prior knowledge of target angles, FDCBF can detect the weak target (3) from 0 to 100 s, whereas the weak target (5) can be vaguely detected from 0 to 10 s. Compared with Figures 11(a) and 11(b), Figure 11(c) shows that RFDCBF can detect Target 3 (Target 5) with a neighboring loud source, i. e., Target 2 (Target 4), from 0 to 100 s, without prior knowledge of target angles.
Thus, we have proven that RFDCBF exhibits excellent performance in weak target detection.
5 Conclusion
In this paper, a focusedanddeconvolved method is proposed to address the wide mainlobe and high sidelobe levels of wideband CBF based on focusing transform and deconvolution. First, the focusedanddeconvolved beam power (FDCBF and RFDCBF) is simulated and evaluated in terms of the mainlobe width, first sidelobe level, and RMSE. Then, DOA estimation is analyzed in terms of the bearing resolution performance and BTRs. Subsequently, antinoise performance is assessed from aspects of the ABL and SSG. Finally, considering the feasibility of practical application, RFDCBF is confirmed to be more suitable for actual data processing without prior knowledge of target angles, compared with FDCBF.
Competing interest Xueli Sheng is an editorial board member for the Journal of Marine Science and Application and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no other competing interests. 
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